Learn Hardcore Icelandic: Eftir samtalið hjá bankanum fer hún til vinstri. - After the conversation at the bank she goes to the left.

Icelandic main clauses are verb-second (V2). Because the time/place phrase Eftir samtalið hjá bankanum is fronted, the finite verb must come next, so we get fer hún. Neutral order without fronting would be: Hún fer til vinstri eftir samtalið hjá bankanum. The pattern Eftir samtalið hjá bankanum hún fer… is ungrammatical.

Which cases are used here, and why?

eftir governs the accusative when it means “after (in time),” so samtalið is accusative (neuter, definite: from samtal → samtalið).
hjá governs the dative, hence bankanum (from banki → dative definite bankanum).
til normally governs the genitive. In the fixed direction phrases til vinstri/til hægri the form is set as vinstri/hægri; you can also see the variant til vinstrar in some styles, but til vinstri is the default.

Why use hjá bankanum instead of í bankanum or við bankann?

hjá = “at/by someone’s place/premises” or “at an institution”: hjá bankanum ≈ “at the bank (branch/premises).”
í = “in/inside”: í bankanum means the conversation was inside the bank.
við with verbs like tala við = “talk to/with (someone)”: Eftir samtalið við bankann = “after the conversation with the bank (as an interlocutor, e.g., on the phone).”
Choose based on what you want to emphasize: location at the premises (hjá), being inside (í), or the interlocutor (við).

Does hjá bankanum modify samtalið or the verb phrase?

As written, hjá bankanum most naturally attaches to samtalið: “the conversation at the bank.” If you meant “After the conversation, she goes past the bank to the left,” you’d typically rephrase to avoid ambiguity, e.g., Eftir samtalið fer hún hjá bankanum og beygir til vinstri (goes past the bank and turns left) or split into two clauses.

Could I say Eftir samtalinu instead of Eftir samtalið?

Not with the meaning “after (the conversation).” When eftir means “after (in time),” it takes the accusative: Eftir samtalið.
With the dative, eftir means “along/according to,” e.g. Hún gengur eftir veginum (“She walks along the road”) or Fara eftir reglum (“follow rules”).

Can I say Eftir samtalið við bankann? How does that differ from hjá bankanum?

Yes.

Eftir samtalið við bankann = after the conversation with the bank (as your interlocutor).
Eftir samtalið hjá bankanum = after the conversation at the bank’s premises.
Pick við if the partner is the bank (phone/chat), hjá/í if you want to highlight the location.

Is fer til vinstri the most idiomatic way to express “turn left,” or should I use beygir?

All are possible, with nuances:

fer til vinstri = “goes/heads to the left” (common in directions).
beygir til vinstri = “turns left” (very common for driving/walking directions).
snýr til vinstri can mean “turns to the left” (more about rotating/turning one’s body or orientation).
For road directions, beygir til vinstri is often the clearest.

Do I need the preposition til before vinstri? Can I say Hún fer vinstri?

You need the preposition. Say Hún fer/beygir til vinstri. Alternatives:

Haltu til vinstri (“keep left”).
Haltu þig vinstramegin (“keep to the left-hand side”).
Bare vinstri without a preposition does not work as a directional complement here.

Why is the present tense fer used? Would past or future be different?

Present is standard in instructions, narratives, and running commentary: it reads like a set of directions. For a past story: Eftir samtalið… fór hún til vinstri. Future with mun is possible but unnecessary in directions: Hún mun fara… sounds like a prediction rather than an instruction.

Where would I place negation or adverbs?

They come after the verb (and subject) in main clauses:

Eftir samtalið hjá bankanum fer hún ekki til vinstri.
Eftir samtalið hjá bankanum fer hún þá beint.
In yes–no questions, the verb goes first: Fer hún til vinstri eftir samtalið?

What’s going on with the definite endings -ið and -num in samtalið and bankanum?

Icelandic uses suffixed definite articles:

samtal (a conversation) → samtalið (the conversation). Neuter singular nominative/accusative share the same form.
banki (a bank) → dative definite bankanum (the bank).
Indefinite versions would be samtal and hjá banka (“at a bank”). Use definite when referring to a specific, known entity.

Is a comma needed after the fronted phrase?

No. Icelandic typically does not use a comma after a fronted adverbial like this. Eftir samtalið hjá bankanum fer hún til vinstri. is the preferred punctuation.